| It is well known that Gaussian hypergeometric functionF(a,b;c;x),complete elliptic inte-grals Κ(r)and ε(r),generalized elliptic integrals Κa(r)and εa(r)are very important special functions.They play an important role in geometric function theory,quasiconformal theo-ry,number theory,physics and other disciplines,and in engineering technology.However,many properties of Gaussian hypergeometric function F(a,b;c;x)rely on the gamma functionΓ(x),psi function φ(x),beta function B(x,y)and the Ramanujan constant R(a,b).As an im-portant special case of Gaussian hypergeometric function F(a,b;c;x),the generalized elliptic integral Κ(r)is one of the most important special functions,and related to the Schwarz-Christoffel transformation,which maps the upper half plane onto a parallelogram.On the other hand,it is a generalization of the complete elliptic integral.That is,when a = 1/2,Κa(r)reduces to Κ(r).Therefore,generalizing some properties of the complete elliptic integrals to the generalized elliptic integrals,benefits the development of related fields such as geo-metric function theory,quasiconformal theory and the theory of generalized Ramanujan equations.The generalized elliptic integral Κa(r)has a close relation with the generalized Grotzsch ring function μa(r),the generalized Hubner upper bound function ma(r),the gen-eralized Hersch-Pfluger distortion function φK(a,r),the generalized Agard distortion func-tion ηK(a,t)and the generalized linear distortion function λ(a,K).When a =1/2,μa(r)and these generalized distortion functions are the important quasiconformal special functionsμ(r),m(r),φK(r),ηK(t)and λ(K),which are not only necessary in the study of the properties of quasiconformal mappings,but also in many other mathematical fields such as number theory[1,41,53,56].In addition,ηK(t)and λ(K)play an important role in the researches of ex-tremal problems of quasiconformal mappings,quasi-regular mappings and quasisymmetric functions[1,50,61].The study of generalized elliptic integral Κa(r)and generalized distortion functions φK(a,r),ηK(a,t),and λ(a,K)is quite important not only in quasiconformal theory and geometric function theory,but also in generalized Ramanujan equations.Therefore,it is quite significant for us to study the properties of the special functions above-mentioned.The ring domains,or abbreviated,rings such as the Grotzsch and the Teichmuller rings are indispensable in the study of quasiconformal theory.For example,the Grotzsch ring function Mn(r)(M2(r)= μ(r)for n = 2)in Rn plays a very important role in the study of n-dimension quasiconformal theory.The properties of the plane Grotzsch ring function μ(r)has been deeply studied and many good results for μ(r)has been known to us.However,relatively speaking,the known results for Mn(r)(n ≥ 3)are much less and roughes than those forμ(r),because of lacking of effective tools for the study of Mn(r).The main purpose of this thesis is to establish different series expansions of beta func-tion B(a,b),to study the relation between R(a,b)and B(a,b),and the analytic properties of certain combination in terms of Κa(r)and some elementary functions,and to obtain sharp inequalities.We shall also substantively improve an inequality satisfied by Mn(r).This thesis consists of the following four chapters:In Chapter 1,we introduce the background of this thesis,and including some needed con-cepts,notations and some known results.In Chapter 2,the different series expansions of the beta function B(x,y)are estab-lished,and the relation between B(a,b)and R(a,b)is revealed.In Chapter 3,we obtain some monotonicity and convexity properties of the generalized elliptic integral Κa(r)by studying those of certain combinations in terms of Κa(r)and some elementary functions,from which some sharp inequalities for Κa(r).In Chapter 4,we study the properties of the Grotzsch ring function Mn(r)= modRG,n(1/r)in the n-dimensional quasiconformal theory.We shall correct an error in[1],and substantively improve the upper bound of f(r)≡ h(r)+ h(r’),where h(r)= r’2Mn(r)Mn(r’)n-1. |
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